Gershon Wolansky, Technion and Indiana University.

"From Mass Transport to the Hamilton-Jacobi equation"

Abstract

Mass Transport, or Optimal transportation, or the Monge-Kantorovich problem, are common names for the problem of transporting a given probability measure to another, at a minimal prescribed cost. The first formulation was given by Monge in the 17th century. A relaxed formulation was introduced by Kantorovich in the first half of the 20 century. This problem became very popular in the last few decades, as applications to probability, diffusion process, economics, geostrophic flow, image recognition and many other fields were discussed in a countless number of papers. I'll review the basic definitions and introduce another interpretation for mass transport in terms of a minimal flow in a configuration space of paths of probability measures. This leads naturally, via a dual formulation, to an Hamilton-Jacobi equation. I'll discuss its solvability and regularity in some general cases.